Caveat:
Like any drummer, Buddy occasionally used "other" equipment, for example when he flew in for an appearance on the "Tonight Show". So, the equipment I'll list is what he used personally in his bands, most of the time. Same goes for "tuning"... Buddy wasn't the type to sit at the drums for hours with a drum key, (and didn't need to) so of course, you may find some recordings with slightly different tunings than listed here.

----------------- Drums

Buddy played a variety of drums, of course, over his career of nearly 70 years. Typically though, they were what we'd think of as "vintage" today... meaning fewer or single plies of thicker maple. He mostly used a 14" X 24" bass drum, although both early and late in his career, he used a 26" bass drum. Mounted tom was a 9" X 13", along with two 16" X 16" floor toms.

----------------- "Tuning" or "Tensioning", as he liked to call it

Beginning in the 1950's until the end of his life, Buddy preferred Remo white-coated "Diplomat" heads on all of the drums. (and yes, they used to be available as bass drun heads too.) These heads are (thankfully!) still available today, so he would probably still use them. The nearest "equivalent" would be the Remo "Ambassodor" heads. Buddy was instrumental in many of Remo's developments, asking for thinner heads for example. (hence the "Diplomat" and "Diplomat M5"... even thinner) He also asked Remo Belli for a way to make his bass drum heads last a little longer, (He used a wood beater) and hence the "dot" was created.

Tom and bass drum tensioning was the same. "Bottom" or "reso"-side heads were tighter, (Musically, when the heads were new, the reso heads were nearly the same pitch, but as the top heads "sink in" (see below) they could be as much as a minor third higher in pitch) Snare drum bottom head was a Remo clear "Diplomat" snare side head, also tensioned tighter than the top head. (musically, a 4th to a 6th higher in pitch) The APPROXIMATE top head (or "batter" side) pitch of Buddy's drums was usually:

In their "new head" configuration, the drums sound bright and crisp, with the large halls Buddy typically played in providing depth and natural "reverb". Buddy would sometimes let the drums "sink in", meaning that over a week or two, the pitches above would (uniformlly) go down in pitch from a step to a minor third. In this range, they start to sound "loose" and "papery". When using "Ambassodor" heads, they don't stretch as much, so the change isn't as extreme.

----------------- Cymbals

He preferred "A. Zildjian" cymbals, (thinner than most people would expect) and they nearly always included the same setup:

Ride cymbals in the 20" to 22" size, from light to medium (brief period in the 1980's when he used much thicker (higher-pitched) rides

Mostly 14" (Light "New Beat" type) hi-hats, but sometimes, 13"

One 18" thin crash (on his left side)

One 18" med-thin crash (on his right side)

Usually a 6" thin splash, but sometimes an 8" thin splash

At the end of his career, he added a (22" "Swish Knocker") China-type cymbal